Why are the rules the way they are?? What determines them to be true, like if you gave someone two pieces of candy and then gave them another two pieces of candy, then you gave that person 4 pieces of candy( not 5, not 4.5, not 4.1). So when would all these exponents be actually used in real life?? What is the practical day to day approach to this stuff?
Also, for step one, why can't you separate them out like I did in my images?? Why are the numbers trapped under the hood sign thing??
I explained to you that the radical is in part a grouping symbol.
What does \(\displaystyle \sqrt[3]{6}\) mean?
It means a number such that \(\displaystyle \sqrt[3]{6} * \sqrt[3]{6} * \sqrt[3]{6} = 6.\)
What does \(\displaystyle \sqrt{6}\) mean?
It means a number such that \(\displaystyle \sqrt{6} * \sqrt{6} = 6.\)
Is it not obvious then that \(\displaystyle 6^{(1/3)} = \sqrt[3]{6} \ne \sqrt{6} = 6^{(1/2)}?\)
They are two different numbers. For a clearer example,
\(\displaystyle \sqrt[3]{64} = 4 \text { because } 4 * 4 * 4 = 64.\)
\(\displaystyle \sqrt{64} = 8 \text { because } 8 * 8 = 64.\)
\(\displaystyle 4 \ne 8.\)
Basically, the rules of exponents are a bunch of definitions. They fit logically together, but they are not discoveries, but creations. Exponents are a convenient notation to express certain kinds of multiplicative relations.